# Questions tagged [triangulated-categories]

A triangulated category is an additive category equipped with the additional structure of an autoequivalence (called the translation functor) and a class of of triangles satisfying certain axioms.

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### How to construct $X \oplus \Sigma X$ from $X \oplus \Sigma X \oplus \Sigma X \oplus \Sigma^2 X$ without splitting an idempotent?

Let $Z$ be an object in a stable (or triangulated/whatever) category $\mathcal C$. I believe it follows from Thomason's theorem (see The classification of triangulated subcategories) that the ...

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### Is the intersection of two compactly generated localizing subcategories still compactly generated?

Suppose we have a compactly generated triangulated category $\mathcal{T}$ such that the subcategory of compact objects $\mathcal{T}^c$ is essentially small. Let us take $\mathcal{A}, \mathcal{B}$ two ...

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### Is the Balmer spectrum of the derived category of the Balmer spectrum of finite spectra the Balmer spectrum of finite spectra?

$\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\SH{SH}\DeclareMathOperator\Mod{Mod}\newcommand{\fin}{\mathrm{fin}}\newcommand{\cc}{\mathrm{c}}$I'm mainly interested in the dynamics of the Balmer ...

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### triangulated hull and thick hull of $\mathcal{O}_X,\dots,\mathcal{O}_X(n)$

Here I am dealing with the difference of triangulated hull and thick hull. Let $\mathcal{D}$ be a triangulated category and $\mathcal{E}\subset\mathcal{D}$ be a collection of objects. The triangulated ...

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### Voevodsky's motives and Deligne's systems of realizations

$\newcommand{\gm}{\mathrm{gm}}$Let $\mathbf{DM}_{\gm}(\mathbb{Q},\mathbb{Z})$ be Voevodsky's category of geometric motives over $\mathbb{Q}$ with coefficients in $\mathbb{Z}$ (e.g. as on p.124 of ...

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### Example: orthogonal complement of an admissible subcategory is not admissible

This question is closely related to
https://mathoverflow.net/questions/190952/is-the-orthogonal-complement-of-an-admissible-subcategory-admissible-itself
Do we have a known example for an admissible ...

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### Find a Morita equivalent finite cell DG category

I am trying to understand the following statement:
Suppose that $\mathcal{E}$ is a pre-triangulated proper DG category with a full exceptional collection. Then $\mathcal{E}$ is Morita equivalent to a ...

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### Cogenerators in a triangulated category

Let $\mathscr{T}$ be a triangulated category and $[1]$ be the shift functor in $\mathscr{T}$ and $X\in \mathscr{T}$ be a cogenerator i.e. if $Hom(Y,X[i])=0,i\in\mathbb{Z}$, then we have $Y=0$. My ...

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### Understanding Balmer spectra

$\DeclareMathOperator\Spec{Spec}\newcommand{\perf}{\mathrm{perf}}\DeclareMathOperator\SHC{SHC}$I have just finished reading the paper "The spectrum of prime ideals in tensor triangulated ...

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### Verdier localisation

$\newcommand{\Perf}{\operatorname{Perf}}$This is a toy example that I want to understand, I will be grateful for any help. Given a ring $R$ and $A=\Perf(R)$ the category of perfect complexes over $R$ ....

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### Yoneda Ext theorem and extensions

Consider the category of chain complexes over a ring $R$.
We can show that $\text{Ext}^1(M, N)$ classifies extensions using the triangulated category structure: the homotopy kernel of a map $N \...

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### Inducing an equivalence of $G$-equivariant categories

Suppose we have an equivalence of triangulated categories $\Phi : \mathcal{A} \to \mathcal{B}$. Let $G$ be a finite group. Are there any methods/conditions for specifying when one has an induced ...

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### On the not so clear relationship between torsion theories and localization for a newcomer

Given an hereditary torsion theory $(\mathcal{T}, \mathcal{F})$ on an abelian category $\mathcal{A}$, how we can relate this to a localization (i.e Ore localization). This is mentioned with not so ...

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### When semi-simple subcategories "extend" to hearts of t-structures?

Let $A$ be a semi-simple abelian subcategory of a triangulated category $C$ that "generates" $A$ (that is, $C$ equals its own smallest triangulated subcategory that is closed under direct ...

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### Any news about equivalences of periodic triangulated or $\infty$-categories?

There is a very old question (October 2009) Equivalence of derived categories which is not Fourier-Mukai which has been bumped by improving links to the literature in one of the answers and attracted ...

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### When is a thick subcategory the preimage of a weak Serre class under a homological functor?

Let $\pi : \mathcal T \to \mathcal A$ be a homological functor from a stable / triangulated category to an abelian category, and let $\mathcal C \subseteq \mathcal A$ be a weak Serre subcategory. Let $...

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### Minus sign in rotated triangles in Triangulated categories

Let $T$ be a triangulated category and
$$ X \xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} X[1]$$
an exact triangle (or distinguished triangle). TR 2 implies that then the two rotated triangles
$$...

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### Terminology: are there any names for "quotients" of cellular towers in stable categories?

A cellular tower in SH or in a "more general stable homotopy category" is a chain of morphisms $\dots X^{(n)}\stackrel{g^n}{\to} X^{(n+1)}\to \dots$ along with some more data and conditions; ...

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### On the definition and an example of silting/tilting subcategories in a triangulated categories according to a paper by Aihara and Iyama

In the paper "Silting mutation in triangulated categories" by Aihara and Iyama, I stumbled upon this nice definition( Definition 2.1) of a tilting/silting subcategory of a triangulated ...

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### construction of $K_0$-group and Karoubian completion

Let $A$ be a ring. The $K_0$ group of $A$ can be defined in most
old fashioned way as the Grothendieck group of the set of isomorphism classes
of its finitely generated projective $R$ modules, ...

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### How to prove a lemma of Rouquier on the dimension of triangulated categories?

In the paper of Rouquier on the dimension of triangulated categories (found here) lemma 3.5 says:
Lemma Let $\mathcal{T}$ be a triangulated category and let $\mathcal{T}_1$ and $\mathcal{T}_2$ be ...

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### Antisymmetric product of complexes

Usually when one has a short exact sequence of bundles,
\begin{eqnarray}
0\rightarrow A \rightarrow B\rightarrow C\rightarrow 0,
\end{eqnarray}
then there is an associated long exact sequence,
\begin{...

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### What is a most natural categorification of a vector space?

Few days ago I became excited when I learned from an answer to Examples of simple vertex operator algebras (VOAs) that
The irreducible modules of the rank $d$ free boson are naturally parametrized by ...

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### What fails in constructing a homotopy category out of candidate triangles in a triangulated category?

Following Neeman's article "New axioms for triangulated categories", for a triangulated category $\mathscr T$ let $CT(\mathscr T)$ denote the category of candidate triangles, i.e. diagrams
\...

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### Rigid, maximal rigid and cluster-tilting objects

Let $\mathcal{D}$ be a $k$-linear, Hom-finite triangulated category with a Serre functor $\mathbb{S}$. An important class of objects in $\mathcal{D}$ are the cluster-tilting objects, which have many ...

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### localizing subcategories of a nice triangulated category

Suppose that $D(A)$ is the derived category of of a ring A. Let $b\in D(A)$ be a compact object and $B$ the localizing subcategory generated by b (having arbitrary coproduct).
Does the inclusion ...

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### Non-uniqueness of $C$ with $f_!(C) = f_*(1_{\mathcal{C}})$

$\newcommand{\Cc}{\mathcal{C}}$
$\newcommand{\Dd}{\mathcal{D}}$
$\newcommand{\Z}{\mathbb{Z}}$
$\newcommand{\Q}{\mathbb{Q}}$
$\newcommand{\tensor}{\otimes}$
$\newcommand{\colim}{\rm colim}$
$\...

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### Which t-structure extend from subcategories of compact objects uniquely?

Let $T$ be a compactly generated triangulated category, that is, $T$ is closed with respect to small coproducts and equals its own smallest triangulated subcategory closed with respect to coproducts ...

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### Comparison: Formal Wirthmüller isomorphism of Fausk-Hu-May vs. Balmer et. al

$\newcommand{\Cc}{\mathcal{C}}$
$\newcommand{\Dd}{\mathcal{D}}$
$\newcommand{\tensor}{\otimes}$
$\DeclareMathOperator{\Sp}{Sp}$
This question is about comparing the approaches for a formal Wirthmüller ...

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### Subcategory of compactly generated triangulated category

Let T be a compactly generated triangulated category and let T' be a localizing subcategory. Is it automatic that T' is comapctly generated by $T^c \cap T'$, where $T^c$ is compact objects of $T$?
...

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### Bousfield localization of triangulated categories:equivalent conditions

In these notes on pages 60-64 Daniel Murfet proves the equivalence of 6 conditions of what it means for the Verdier quotient to be Bousfield localization. I, however, do not understand certain steps ...

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### Semi-stable bundles as a heart of a t-structure

Given an algebraic curve, the category of semi-stable vector bundles of fixed slope forms an abelian category see here page 2. This is surprising as the vector bundles themselves are not an abelian ...

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### Magic behind idempotent-complete categories a.k.a. why (sometimes) be Karoubian is sexier than be Abelian

It is well know that Karoubian categories (also called idempotent-complete categories) are living between additive and
Abelian categories. While one of the most famous advantages to
work with ...

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### 3x3 lemma in triangulated categories

I am currently reading Le Stum's Rigid Cohomology and have encountered the following passage (proof of Proposition 5.2.16):
The deduction made here seems to be purely "triangulated category-...

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### Are functor categories with triangulated codomains themselves triangulated?

I'm fairly confident that the following assertion is true (but I will confess that I did not verify the octahedral axiom yet):
Let $T$ be a triangulated category and $C$ any category (let's say small ...

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### Higher version of 3x3 lemma

Consider a category of complexes over some additive category, and suppose we have a square that commutes up to homotopy. Then, one can consider mapping cones over all edges of the square, and obtain a ...

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### Questions about $\text{Perf}(A)$ of dg algebra $A$

[ALEXEY ELAGIN AND VALERY A. LUNTS, p.4.] Recall that triangulated category $\text{Perf}(A)$ is defined as the full thick triangulated subcategory of $D(A)$ generated by the dg $A$-module $A$.
[...

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### $T$ is a generator of $\mathcal{T}$ iff. the ideal of $T$-ghost maps is contained in the Jacobson radical of $\mathcal{T}$

Definitions
Let $\mathcal{T}$ be a triangulated category with translation functor $\Sigma$.
Generator: An object $T$ of $\mathcal{T}$ is a generator of $\mathcal{T}$ if $\mathcal{T}(\Sigma^n T,A)=0$ ...

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### Computing Ext groups in a functor stable $\infty$-category

Let $I$ be a small category and $\mathcal{D}=D^b_\infty(\mathbb{Z})$ the bounded derived $\infty$-category of abelian groups. Consider the $\infty$-category $\mathcal{C}:=\mathrm{Fun}(I,\mathcal{D})$. ...

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### Can morphisms of Mayer-Vietoris triangles be completed into a $3\times 3$ square?

Let $(X,\mathcal{O}_X)$ be a topological ringed space and $(U,V)$ be an open covering of $X$ (i.e. $U$ and $V$ are two open subsets of $X$ such that $U\cup V=X$). Let $\mathcal{F}$ and $\mathcal{F}'$ ...

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### Cone of a morphism of complexes that are concentrated in degree $0$ and $1$

Let $R$ be a ring and $f:A\to A'$ and $g:B\to B'$ be morphisms of $R$-modules. Let $h:C_{\bullet}\to C_{\bullet}'$ be a morphism of $R$-module complexes fitting in a morphism of distinguished ...

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### Smallness condition for augmented algebras

I'm not sure this question is research level question. Sorry in advance.
Hypothesis
$k$ is a commutative ring.
$A$ is an augmented $k$-algebra.
$A^e$ is defined as the $k$-algebra $A\otimes_{k}A^{op}$...

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### Who introduced the heart ($\mathcal{C}^\heartsuit$) notation in the context of $t$-structures on triangulated categories?

In the context of $t$-structures
([Wikipedia],
[nLab],
[Notes I],
[Notes II],
[HA, Definition 1.2.1.11)],
[BBD, Définition 1.3.1]),
one often writes $\mathcal{C}^\heartsuit$ for the heart of a ...

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### Homotopy equivalences and Mapping Cones

Edit: Version 2:
Suppose that $A,B,C$ are chain complexes and $f: A \rightarrow B$ is a chain map. Suppose that there is a homotopy equivalence
$$ \text{Cone}(f: A \rightarrow B) \simeq C.$$
The chain ...

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### "Universal" triangulated category

Let $\mathcal{C}$ be some category. One way to map this category into a triangulated category is to take the category of simplicial objects $s\mathcal{C}$ (which is an $\infty$-category), take its ...

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### Proof in the realm of $\infty$ categories

I have recently started learning the language of $\infty$-categories. My approach is more to their use, rather than for their own sake. For this reason, as I feel I reached a good understanding of the ...

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### Does localization at quasi-isomorphisms imply homotopy invariance?

Usually, the derived category of some abelian category $A$ (I'm happy already with $A$-mod) is defined first taking chain complexes up to homotopy, and then localize at quasi-isomorphisms.
My question ...

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### Admissibility of intersection of subcategories

Let $\mathscr{T}$ be a triangulated category, and $\mathscr{A}$ be a right admissible subcategory, which means that $i_{\mathscr{A}} : \mathscr{A} \rightarrow \mathscr{T}$ has a right adjoint $i_{\...

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### idea and intuition behind triangulated category

I have some trouble in understanding the significance of some axiom of triangulated category.
If someone could explain to me each axiom with some intuition, and explain to me the intuition behind the ...

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### Why are Serre functors always exact?

Let $k$ be a field and $\mathcal{T}$ be a $k$-linear triangulated category with finite dimensional spaces of morphisms. Bondal and Kapranov proved that every Serre functor on $\mathcal{T}$ is exact (...